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Lecture 6 - System structures for implementation presents the following content: Block diagram representation of computational structures, signal flow graph description, basic structures for IIR systems, transposed forms, basic structures for FIR systems.
Digital Signal Processing, Fall 2006 Lecture 6: System structures for implementation Zheng-Hua Tan Department of Electronic Systems Aalborg University, Denmark zt@kom.aau.dk Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Course at a glance MM1 Discrete-time signals and systems MM2 Fourier-domain representation Sampling and reconstruction MM4 z-transform MM3 DFT/FFT System System analysis System structure MM6 MM5 Filter design MM7, MM8 MM9, MM10 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 System implementation LTI systems with rational system function e.g b0 + b1 z −1 H ( z) = , − az −1 Impulse response | z |>| a | h[n] = b0 a nu[n] + b1a n −1u[n − 1] Linear constant-coefficient difference equation y[n] − ay[n − 1] = b0 x[n] + b1 x[n − 1] Three equivalent representations! How to implement, i.e convert to an algorithm or structure? Digital Signal Processing, VI, Zheng-Hua Tan, 2006 System implementation The input-output transformation x[n] Ỉ y[n] can be computed in different ways – each way is called an implementation An implementation is a specific description of its internal computational structure The choice of an implementation concerns with computational requirements memory requirements, effects of finite-precision, and so on Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Part I: Block diagram representation Block diagram representation of computational structures Signal flow graph description Basic structures for IIR systems Transposed forms Basic structures for FIR systems Digital Signal Processing, VI, Zheng-Hua Tan, 2006 System implementation Impulse response h[n] = b0 a nu[n] + b1a n −1u[n − 1] y[n] = x[n] * h[n] is infinite-duration, impossible to implement in this way However, linear constant-coefficient difference equation provides a means for recursive computation of the output y[n] − ay[n − 1] = b0 x[n] + b1 x[n − 1] y[n] = ay[n − 1] + b0 x[n] + b1 x[n − 1] Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Basic elements Implementation based on the recurrence formula derived from difference equation requires adders y[n] = ay[n − 1] + b0 x[n] + b1 x[n − 1] multipliers memory for storing delayed sequence values Fig 6.1 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Example of block diagram representation A second-order difference equation y[n] = a1 y[n − 1] + a2 y[n − 2] + b0 x[n] H ( z) = b0 − a1 z −1 − a2 z − Demonstrates the complexity, the steps, the amount of resources required Digital Signal Processing, VI, Zheng-Hua Tan, 2006 General Nth-order difference equation N M y[n] = ∑ ak y[n − k ] + ∑ bk x[n − k ] k =1 k =0 M H ( z) = v[n] ∑b z k =0 N −k k − ∑ ak z − k k =1 A cascade of two systems! X[n]Ỉv[n], v[n]Ỉy[n] Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Rearrangement of block diagram 10 A block diagram can be rearranged in many ways without changing overal function, e.g by reversing the order of the two cascaded systems Digital Signal Processing, VI, Zheng-Hua Tan, 2006 System function decomposition v[n] M H ( z) = ∑b z k =0 N −k k − ∑ ak z − k k =1 ⎛ ⎞ ⎜ ⎟ M ⎜ ⎟⎛⎜ b z − k ⎞⎟ = H ( z ) H1 ( z ) = ∑k ⎠ N ⎜ − k ⎟⎝ k = ⎜ − ∑ ak z ⎟ ⎝ k =1 ⎠ ⎛ ⎞ ⎜ ⎟ M ⎛ − k ⎞⎜ ⎟ = H1 ( z ) H ( z ) = ⎜ ∑ bk z ⎟ N ⎜ ⎝ k =0 ⎠ − a z −k ⎟ ⎜ ∑ k ⎟ ⎝ k =1 ⎠ w[n] 11 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 In the time domain N M k =1 k =0 y[n] = ∑ ak y[n − k ] + ∑ bk x[n − k ] M ⎧ v [ n ] bk x[n − k ] = ∑ ⎪⎪ k =0 ⎨ N ⎪ y[n] = ∑ ak y[n − k ] + v[n] ⎪⎩ k =1 N ⎧ w [ n ] ak w[n − k ] + x[n] = ∑ ⎪⎪ k =1 ⎨ M ⎪ y[n] = ∑ bk w[n − k ] ⎪⎩ k =0 12 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Minimum delay implementation One big difference btw the two implementations concerns the number of delay elements N +M max( N , M ) 13 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Direct form I and II Direct form I as shown in Fig 6.3 Direct form II or canonic direct form as shown in Fig 6.5 14 A direct realization of the difference equation There is a direct link between the system function (difference equation) and the block diagram Digital Signal Processing, VI, Zheng-Hua Tan, 2006 An example 15 Direct form I and direct form II implementation + z −1 H ( z) = − 1.5 z −1 + 0.9 z − Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Part II: Signal flow graph description 16 Block diagram representation of computational structures Signal flow graph description Basic structures for IIR systems Transposed forms Basic structures for FIR systems Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Signal flow graph (SFG) As an alternative to block diagrams with a few notational differences A network of directed branches connecting nodes variable 17 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Signal flow graph 18 Nodes in SFG represent both branching points and adders (depending on the number of incoming branches), while in the diagram a special symbol is used for adders and a node has only one incoming branch SGF is simpler to draw Digital Signal Processing, VI, Zheng-Hua Tan, 2006 From flow graph to system function Fig 6.12 w1[n] = w4 [n] − x[n] w2 [n] = αw1[n] w3 [n] = w2 [n] + x[n] w4 [n] = w3 [n − 1] y[n] = w2 [n] + w4 [n] Not a direct form, cannot obtain H(z) by inspection But can write an equation for each node w4 [ n ] = w3 [ n − 1] involve feedback, difficult to solve 19 By z-transform Æ linear equations Digital Signal Processing, VI, Zheng-Hua Tan, 2006 From flow graph to system function ⎧W1 ( z ) = W4 ( z ) − X ( z ) ⎪ ⎪W2 ( z ) = αW1 ( z ) ⎪ ⎨W3 ( z ) = W2 ( z ) + X ( z ) ⎪ −1 ⎪W4 ( z ) = z W3 ( z ) ⎪⎩Y ( z ) = W2 ( z ) + W4 ( z ) ⎧W2 ( z ) = α (W4 ( z ) − X ( z )) ⎪ ⎪ ⎨ −1 ⎪W4 ( z ) = z (W2 ( z ) + X ( z )) ⎪Y ( z ) = W ( z ) + W ( z ) ⎩ ⎛ z −1 − α ⎞ ⎟ X ( z) Y ( z ) = ⎜⎜ −1 ⎟ ⎝ − αz ⎠ z −1 − α If α is real, the system is ? H ( z) = − αz −1 Causal! h[n] = α n −1u[n − 1] − α n +1u[n] 20 All-pass Digital Signal Processing, VI, Zheng-Hua Tan, 2006 10 From flow graph to system function Fig 6.13 Fig 6.12 Different implementations, different amounts of computational resources 21 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Part III: Basic structures for IIR systems 22 Block diagram representation of computational structures Signal flow graph description Basic structures for IIR systems Transposed forms Basic structures for FIR systems Digital Signal Processing, VI, Zheng-Hua Tan, 2006 11 Direct form I N k =1 M M y[n] = ∑ ak y[n − k ] + ∑ bk x[n − k ] k =0 H ( z) = ∑b z k =0 N −k k − ∑ ak z − k k =1 23 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Direct form II N y[n] = ∑ ak y[n − k ] + ∑ bk x[n − k ] k =1 M M k =0 H ( z) = ∑b z k =0 N −k k − ∑ ak z − k k =1 24 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 12 Example H ( z) = 25 + z −1 + z −2 − 0.752 z −1 + 0.125 z − Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Cascade form Factor the numerator and denominator polynomials M H ( z) = ∑ bk z − k k =0 N − ∑ ak z − k k =1 =A M1 M2 k =1 N1 k =1 N2 k =1 k =1 ∏ (1 − f k z −1 )∏ (1 − g k z −1 )(1 − g k* z −1 ) ∏ (1 − ck z −1 )∏ (1 − d k z −1 )(1 − d k* z −1 ) b + b z −1 + b2 k z −2 H ( z ) = ∏ k 1k −1 − a2 k z − k =1 − a1k z Ns 26 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 13 An example: from 2nd-order to 1st-order cascade H ( z) = 27 + z −1 + z −2 (1 + z −1 ) ⋅ (1 + z −1 ) = − 0.752 z −1 + 0.125 z − (1 − 0.5 z −1 ) ⋅ (1 − 0.25 z −1 ) Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Parallel form by partial fraction expansion Np N1 Ak −1 k =1 − c k z H ( z) = ∑ Ck z −k + ∑ k =0 N2 Bk (1 − e k z −1 ) k =1 (1 − d k z −1 )(1 − d *k z −1 ) +∑ 28 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 14 Feedback in IIR systems Feedback loop: a closed path Necessary but not sufficient condition for IIR system (Feedback introduced poles could be cancelled by zeros) a nu[n] H ( z) = y[n] = ay[n] + x[n] y[n] = x[n] /(1 − a) 29 − a z −2 = + az −1 −1 − az All loops must contain at least one unit delay element Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Part IV: Transposed forms 30 Block diagram representation of computational structures Signal flow graph description Basic structures for IIR systems Transposed forms Basic structures for FIR systems Digital Signal Processing, VI, Zheng-Hua Tan, 2006 15 Transposed form for a first-order system Flow graph reversal or transposition also provides alternatives: reversing the directions of all branches and reversing the input and output Resulting in same H(z) H ( z) = 31 1 − az −1 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Transposed direct form II and direct form II The transposed direct form II implements the zeros first and then the poles, being important effect for finite-precision existing 32 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 16 Part V: Basic structures for FIR systems 33 Block diagram representation of computational structures Signal flow graph description Basic structures for IIR systems Transposed forms Basic structures for FIR systems Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Direct form 34 So far, system function has both poles and zeros FIR systems as a special case Causal FIR system function has only zeros (except for poles as z=0) M ⎧b , n = 0,1, , M y[n] = ∑ bk x[n − k ] h[n] = ⎨ n k =0 ⎩0, otherwise Form I and form II are the same Digital Signal Processing, VI, Zheng-Hua Tan, 2006 17 Cascade form Factoring the polynomial system function M H ( z ) = ∑ h[n]z n =0 35 −n Ms = ∏ (b0 k + b1k z −1 + b2 k z − ) k =1 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Linear-phase FIR systems Generalized linear-phase system H (e jω ) = A(e jω )e − jωα + jβ A(e jω ) is a real function of ω , α and β are real constants Causal FIR systems have generalized linear-phase if h[n] satisfies the symmetry condition h[ M − n] = h[n], n = 0,1, , M or h[ M − n] = −h[n], n = 0,1, , M 36 M y[n] = ∑ bk x[n − k ] k =0 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 18 Linear-phase FIR systems if M is an even integer M y[n] = ∑ h[k ]x[n − k ] k =0 = M / −1 ∑ h[k ]x[n − k ] + h[k / M ]x[n − M / 2] + k =0 = M ∑ h[k ]x[n − k ] k = M / +1 M / −1 M / −1 k =0 k =0 ∑ h[k ]x[n − k ] + h[k / M ]x[n − M / 2] + ∑ h[M − k ]x[n − M + k ] if h[ M − n] = h[n] y[n] = M / −1 ∑ h[k ]( x[n − k ] + x[n − M + k ]) + h[k / M ]x[n − M / 2] k =0 37 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Linear phase FIR systems M is an even integer and h[M-n]=h[n] y[n] = M / −1 ∑ h[k ]( x[n − k ] + x[n − M + k ]) + h[k / M ]x[n − M / 2] k =0 38 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 19 Discussions 39 Implementation of FIR and IIR systems Use signal block diagram flow graph representation to show the computational structures Although two structures may have equivalent inputoutput charateristics for infinite-precision represenations of coefficients and variables, they may have dramatically different behaviour when the numerical precision is limited Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Summary 40 Block diagram representation of computational structures Signal flow graph description Basic structures for IIR systems Transposed forms Basic structures for FIR systems Digital Signal Processing, VI, Zheng-Hua Tan, 2006 20 Course at a glance MM1 Discrete-time signals and systems MM2 Fourier-domain representation Sampling and reconstruction MM4 z-transform MM3 41 DFT/FFT System System analysis System structure MM6 MM5 Filter design MM7, MM8 MM9, MM10 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 21 ... diagram Digital Signal Processing, VI, Zheng-Hua Tan, 20 06 An example 15 Direct form I and direct form II implementation + z −1 H ( z) = − 1.5 z −1 + 0.9 z − Digital Signal Processing, VI, Zheng-Hua. .. VI, Zheng-Hua Tan, 20 06 Direct form II N y[n] = ∑ ak y[n − k ] + ∑ bk x[n − k ] k =1 M M k =0 H ( z) = ∑b z k =0 N −k k − ∑ ak z − k k =1 24 Digital Signal Processing, VI, Zheng-Hua Tan, 20 06 12... systems Digital Signal Processing, VI, Zheng-Hua Tan, 20 06 20 Course at a glance MM1 Discrete-time signals and systems MM2 Fourier-domain representation Sampling and reconstruction MM4 z-transform